October 2018, 12(5): 1199-1217. doi: 10.3934/ipi.2018050

Retinex based on exponent-type total variation scheme

1. 

Center for Applied Mathematics, Tianjin University, Tianjin, China

2. 

School of Mathematics and Statistics, and Laboratory of Data Analysis Technology, Henan University, Kaifeng 475004, China

3. 

Center for Applied Mathematics, Tianjin University, Tianjin, China

* Corresponding author: Yuping Duan. yuping.duan@tju.edu.cn

Received  August 2017 Revised  March 2018 Published  July 2018

Fund Project: The work is supported by the 1000 Talents Program for Young Scientists of China, the Ministry of Science and Technology of China ("863" Program: 2015AA020101), and NSFC 11701418 and 11526208. Dr. Z.-F. Pang was partially supported by National Basic Research Program of China (973 Program No.2015CB856003) and NSFC (Nos.U1304610 and 11401170), and also gratefully acknowledges financial support from China Scholarship Council(CSC) as a research scholar to visit the University of Liverpool from August 2017 to August 2018

Retinex theory deals with compensation for illumination effects in images, which has a number of applications including Retinex illusion, medical image intensity inhomogeneity and color image shadow effect etc.. Such ill-posed problem has been studied by researchers for decades. However, most exiting methods paid little attention to the noises contained in the images and lost effectiveness when the noises increase. The main aim of this paper is to present a general Retinex model to effectively and robustly restore images degenerated by both illusion and noises. We propose a novel variational model by incorporating appropriate regularization technique for the reflectance component and illumination component accordingly. Although the proposed model is non-convex, we prove the existence of the minimizers theoretically. Furthermore, we design a fast and efficient alternating minimization algorithm for the proposed model, where all subproblems have the closed-form solutions. Applications of the algorithm to various gray images and color images with noises of different distributions yield promising results.

Citation: Lu Liu, Zhi-Feng Pang, Yuping Duan. Retinex based on exponent-type total variation scheme. Inverse Problems & Imaging, 2018, 12 (5) : 1199-1217. doi: 10.3934/ipi.2018050
References:
[1]

J.-F. Aujol and A. Chambolle, Dual norms and image decomposition models, International Journal of Computer Vision, 63 (2005), 85-104. doi: 10.1007/s11263-005-4948-3.

[2]

J.-F. AujolG. GilboaT. Chan and S. Osher, Structure-texture image decomposition-modeling, algorithms, and parameter selection, International Journal of Computer Vision, 67 (2006), 111-136. doi: 10.1007/s11263-006-4331-z.

[3]

M. Benning, F. Knoll, C.-B. Schönlieb and T. Valkonen, Preconditioned admm with nonlinear operator constraint, in IFIP Conference on System Modeling and Optimization, (2015), 117–126. doi: 10.1007/978-3-319-55795-3_10.

[4]

M. BertalmíoV. Caselles and E. Provenzi, Issues about retinex theory and contrast enhancement, International Journal of Computer Vision, 83 (2009), 101-119. doi: 10.1007/s11263-009-0221-5.

[5]

M. BertalmíoV. CasellesE. Provenzi and A. Rizzi, Perceptual color correction through variational techniques, IEEE Transactions on Image Processing, 16 (2007), 1058-1072. doi: 10.1109/TIP.2007.891777.

[6]

A. Bovik, Handbook of Image and Video Processing, Academic Press, 2000.

[7]

A. Chambolle and P.-L. Lions, Image recovery via total variation minimization and related problems, Numerische Mathematik, 76 (1997), 167-188. doi: 10.1007/s002110050258.

[8]

H. Chang, M. K. Ng, W. Wang and T. Zeng, Retinex image enhancement via a learned dictionary, Optical Engineering, 54 (2015), 013107. doi: 10.1117/1.OE.54.1.013107.

[9]

H. ChangW. HuangC. WuS. HuangC. GuanS. SekarK. K. Bhakoo and Y. Duan, A new variational method for bias correction and its applications to rodent brain extraction, IEEE Transactions on Medical Imaging, 36 (2017), 721-733. doi: 10.1109/TMI.2016.2636026.

[10]

T. J. Cooper and F. A. Baqai, Analysis and extensions of the frankle-mccann retinex algorithm, Journal of Electronic Imaging, 13 (2004), 85-93. doi: 10.1117/1.1636182.

[11]

Y. Duan, H. Chang, W. Huang and J. Zhou, Simultaneous bias correction and image segmentation via L0 regularized mumford-shah model, in 2014 IEEE International Conference on Image Processing (ICIP), (2014), 6–10. doi: 10.1109/ICIP.2014.7025000.

[12]

Y. DuanH. ChangW. HuangJ. ZhouZ. Lu and C. Wu, The $L_0$ regularized mumford-shah model for bias correction and segmentation of medical images, IEEE Transactions on Image Processing, 24 (2015), 3927-3938. doi: 10.1109/TIP.2015.2451957.

[13]

M. Elad, Retinex by two bilateral filters, in International Conference on Scale-Space Theories in Computer Vision, Springer, Berlin, (2005), 217–229. doi: 10.1007/11408031_19.

[14]

O. Faugeras, Digital color image processing within the framework of a human visual model, IEEE Transactions on Acoustics, Speech, and Signal Processing, 27 (1979), 380-393. doi: 10.1109/TASSP.1979.1163262.

[15]

T. Goldstein and S. Osher, The split bregman method for L1-regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343. doi: 10.1137/080725891.

[16]

Y.-M. HuangM. K. Ng and Y.-W. Wen, A new total variation method for multiplicative noise removal, SIAM Journal on Imaging Sciences, 2 (2009), 20-40. doi: 10.1137/080712593.

[17]

D. J. JobsonZ.-U. Rahman and G. A. Woodell, Properties and performance of a center/surround retinex, IEEE Transactions on Image Processing, 6 (1997), 451-462. doi: 10.1109/83.557356.

[18]

R. KimmelM. EladD. ShakedR. Keshet and I. Sobel, A variational framework for retinex, International Journal of Computer Vision, 52 (2003), 7-23. doi: 10.1023/A:1022314423998.

[19]

E. H. Land and J. J. McCann, Lightness and retinex theory, Journal of the Optical Society of America, 61 (1971), 1-11. doi: 10.1364/JOSA.61.000001.

[20]

E. H. Land, Recent advances in retinex theory and some implications for cortical computations: color vision and the natural image, Proceedings of the National Academy of Sciences, 80 (1983), 5163-5169. doi: 10.1073/pnas.80.16.5163.

[21]

E. H. Land, An alternative technique for the computation of the designator in the retinex theory of color vision, Proceedings of the National Academy of Sciences, 83 (1986), 3078-3080. doi: 10.1073/pnas.83.10.3078.

[22]

T. LeR. Chartrand and T. J. Asaki, A variational approach to reconstructing images corrupted by poisson noise, Journal of Mathematical Imaging and Vision, 27 (2007), 257-263. doi: 10.1007/s10851-007-0652-y.

[23]

J. Liang and X. Zhang, Retinex by higher order total variation ${L}^1$ decomposition, Journal of Mathematical Imaging and Vision, 52 (2015), 345-355. doi: 10.1007/s10851-015-0568-x.

[24]

L. Liu, Z.-F. Pang and Y. Duan, A novel variational model for retinex in presence of severe noises, in 2017 IEEE International Conference on Image Processing (ICIP), (2017), 3490–3494. doi: 10.1109/ICIP.2017.8296931.

[25]

W. Ma and S. Osher, A tv bregman iterative model of retinex theory, Inverse Problems and Imaging, 6 (2012), 697-708. doi: 10.3934/ipi.2012.6.697.

[26]

J. McCann, Lessons learned from mondrians applied to real images and color gamuts, in Proceedings of the IST/SID 7th Color Imaging Conference, (1999), 1–8.

[27]

J. M. MorelA. B. Petro and C. Sbert, A pde formalization of retinex theory, IEEE Transactions on Image Processing, 19 (2010), 2825-2837. doi: 10.1109/TIP.2010.2049239.

[28]

M. K. Ng and W. Wang, A total variation model for retinex, SIAM Journal on Imaging Sciences, 4 (2011), 345-365. doi: 10.1137/100806588.

[29]

R. Palma-AmestoyE. ProvenziM. Bertalmío and V. Caselles, A perceptually inspired variational framework for color enhancement, IEEE Transactions on Pattern Analysis and Machine Intelligence, 31 (2009), 458-474. doi: 10.1109/TPAMI.2008.86.

[30]

K. Papafitsoros and C.-B. Schönlieb, A combined first and second order variational approach for image reconstruction, Journal of Mathematical Imaging and Vision, 48 (2014), 308-338. doi: 10.1007/s10851-013-0445-4.

[31]

E. ProvenziD. MariniL. De Carli and A. Rizzi, Mathematical definition and analysis of the retinex algorithm, Journal of the Optical Society of America A, 22 (2005), 2613-2621. doi: 10.1364/JOSAA.22.002613.

[32]

L. I. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.

[33]

Y. WangJ. YangW. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction, SIAM Journal on Imaging Sciences, 1 (2008), 248-272. doi: 10.1137/080724265.

[34]

C. Wu and X.-C. Tai, Augmented lagrangian method, dual methods, and split bregman iteration for rof, vectorial tv, and high order models, SIAM Journal on Imaging Sciences, 3 (2010), 300-339. doi: 10.1137/090767558.

[35]

C. WuJ. Zhang and X.-C. Tai, Augmented lagrangian method for total variation restoration with non-quadratic fidelity, Inverse Problems and Imaging, 5 (2011), 237-261. doi: 10.3934/ipi.2011.5.237.

[36]

X. ZhangM. Burger and S. Osher, A unified primal-dual algorithm framework based on bregman iteration, Journal of Scientific Computing, 46 (2011), 20-46. doi: 10.1007/s10915-010-9408-8.

[37]

D. ZossoG. Tran and S. Osher, Non-local retinex--a unifying framework and beyond, SIAM Journal on Imaging Sciences, 8 (2015), 787-826. doi: 10.1137/140972664.

show all references

References:
[1]

J.-F. Aujol and A. Chambolle, Dual norms and image decomposition models, International Journal of Computer Vision, 63 (2005), 85-104. doi: 10.1007/s11263-005-4948-3.

[2]

J.-F. AujolG. GilboaT. Chan and S. Osher, Structure-texture image decomposition-modeling, algorithms, and parameter selection, International Journal of Computer Vision, 67 (2006), 111-136. doi: 10.1007/s11263-006-4331-z.

[3]

M. Benning, F. Knoll, C.-B. Schönlieb and T. Valkonen, Preconditioned admm with nonlinear operator constraint, in IFIP Conference on System Modeling and Optimization, (2015), 117–126. doi: 10.1007/978-3-319-55795-3_10.

[4]

M. BertalmíoV. Caselles and E. Provenzi, Issues about retinex theory and contrast enhancement, International Journal of Computer Vision, 83 (2009), 101-119. doi: 10.1007/s11263-009-0221-5.

[5]

M. BertalmíoV. CasellesE. Provenzi and A. Rizzi, Perceptual color correction through variational techniques, IEEE Transactions on Image Processing, 16 (2007), 1058-1072. doi: 10.1109/TIP.2007.891777.

[6]

A. Bovik, Handbook of Image and Video Processing, Academic Press, 2000.

[7]

A. Chambolle and P.-L. Lions, Image recovery via total variation minimization and related problems, Numerische Mathematik, 76 (1997), 167-188. doi: 10.1007/s002110050258.

[8]

H. Chang, M. K. Ng, W. Wang and T. Zeng, Retinex image enhancement via a learned dictionary, Optical Engineering, 54 (2015), 013107. doi: 10.1117/1.OE.54.1.013107.

[9]

H. ChangW. HuangC. WuS. HuangC. GuanS. SekarK. K. Bhakoo and Y. Duan, A new variational method for bias correction and its applications to rodent brain extraction, IEEE Transactions on Medical Imaging, 36 (2017), 721-733. doi: 10.1109/TMI.2016.2636026.

[10]

T. J. Cooper and F. A. Baqai, Analysis and extensions of the frankle-mccann retinex algorithm, Journal of Electronic Imaging, 13 (2004), 85-93. doi: 10.1117/1.1636182.

[11]

Y. Duan, H. Chang, W. Huang and J. Zhou, Simultaneous bias correction and image segmentation via L0 regularized mumford-shah model, in 2014 IEEE International Conference on Image Processing (ICIP), (2014), 6–10. doi: 10.1109/ICIP.2014.7025000.

[12]

Y. DuanH. ChangW. HuangJ. ZhouZ. Lu and C. Wu, The $L_0$ regularized mumford-shah model for bias correction and segmentation of medical images, IEEE Transactions on Image Processing, 24 (2015), 3927-3938. doi: 10.1109/TIP.2015.2451957.

[13]

M. Elad, Retinex by two bilateral filters, in International Conference on Scale-Space Theories in Computer Vision, Springer, Berlin, (2005), 217–229. doi: 10.1007/11408031_19.

[14]

O. Faugeras, Digital color image processing within the framework of a human visual model, IEEE Transactions on Acoustics, Speech, and Signal Processing, 27 (1979), 380-393. doi: 10.1109/TASSP.1979.1163262.

[15]

T. Goldstein and S. Osher, The split bregman method for L1-regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343. doi: 10.1137/080725891.

[16]

Y.-M. HuangM. K. Ng and Y.-W. Wen, A new total variation method for multiplicative noise removal, SIAM Journal on Imaging Sciences, 2 (2009), 20-40. doi: 10.1137/080712593.

[17]

D. J. JobsonZ.-U. Rahman and G. A. Woodell, Properties and performance of a center/surround retinex, IEEE Transactions on Image Processing, 6 (1997), 451-462. doi: 10.1109/83.557356.

[18]

R. KimmelM. EladD. ShakedR. Keshet and I. Sobel, A variational framework for retinex, International Journal of Computer Vision, 52 (2003), 7-23. doi: 10.1023/A:1022314423998.

[19]

E. H. Land and J. J. McCann, Lightness and retinex theory, Journal of the Optical Society of America, 61 (1971), 1-11. doi: 10.1364/JOSA.61.000001.

[20]

E. H. Land, Recent advances in retinex theory and some implications for cortical computations: color vision and the natural image, Proceedings of the National Academy of Sciences, 80 (1983), 5163-5169. doi: 10.1073/pnas.80.16.5163.

[21]

E. H. Land, An alternative technique for the computation of the designator in the retinex theory of color vision, Proceedings of the National Academy of Sciences, 83 (1986), 3078-3080. doi: 10.1073/pnas.83.10.3078.

[22]

T. LeR. Chartrand and T. J. Asaki, A variational approach to reconstructing images corrupted by poisson noise, Journal of Mathematical Imaging and Vision, 27 (2007), 257-263. doi: 10.1007/s10851-007-0652-y.

[23]

J. Liang and X. Zhang, Retinex by higher order total variation ${L}^1$ decomposition, Journal of Mathematical Imaging and Vision, 52 (2015), 345-355. doi: 10.1007/s10851-015-0568-x.

[24]

L. Liu, Z.-F. Pang and Y. Duan, A novel variational model for retinex in presence of severe noises, in 2017 IEEE International Conference on Image Processing (ICIP), (2017), 3490–3494. doi: 10.1109/ICIP.2017.8296931.

[25]

W. Ma and S. Osher, A tv bregman iterative model of retinex theory, Inverse Problems and Imaging, 6 (2012), 697-708. doi: 10.3934/ipi.2012.6.697.

[26]

J. McCann, Lessons learned from mondrians applied to real images and color gamuts, in Proceedings of the IST/SID 7th Color Imaging Conference, (1999), 1–8.

[27]

J. M. MorelA. B. Petro and C. Sbert, A pde formalization of retinex theory, IEEE Transactions on Image Processing, 19 (2010), 2825-2837. doi: 10.1109/TIP.2010.2049239.

[28]

M. K. Ng and W. Wang, A total variation model for retinex, SIAM Journal on Imaging Sciences, 4 (2011), 345-365. doi: 10.1137/100806588.

[29]

R. Palma-AmestoyE. ProvenziM. Bertalmío and V. Caselles, A perceptually inspired variational framework for color enhancement, IEEE Transactions on Pattern Analysis and Machine Intelligence, 31 (2009), 458-474. doi: 10.1109/TPAMI.2008.86.

[30]

K. Papafitsoros and C.-B. Schönlieb, A combined first and second order variational approach for image reconstruction, Journal of Mathematical Imaging and Vision, 48 (2014), 308-338. doi: 10.1007/s10851-013-0445-4.

[31]

E. ProvenziD. MariniL. De Carli and A. Rizzi, Mathematical definition and analysis of the retinex algorithm, Journal of the Optical Society of America A, 22 (2005), 2613-2621. doi: 10.1364/JOSAA.22.002613.

[32]

L. I. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.

[33]

Y. WangJ. YangW. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction, SIAM Journal on Imaging Sciences, 1 (2008), 248-272. doi: 10.1137/080724265.

[34]

C. Wu and X.-C. Tai, Augmented lagrangian method, dual methods, and split bregman iteration for rof, vectorial tv, and high order models, SIAM Journal on Imaging Sciences, 3 (2010), 300-339. doi: 10.1137/090767558.

[35]

C. WuJ. Zhang and X.-C. Tai, Augmented lagrangian method for total variation restoration with non-quadratic fidelity, Inverse Problems and Imaging, 5 (2011), 237-261. doi: 10.3934/ipi.2011.5.237.

[36]

X. ZhangM. Burger and S. Osher, A unified primal-dual algorithm framework based on bregman iteration, Journal of Scientific Computing, 46 (2011), 20-46. doi: 10.1007/s10915-010-9408-8.

[37]

D. ZossoG. Tran and S. Osher, Non-local retinex--a unifying framework and beyond, SIAM Journal on Imaging Sciences, 8 (2015), 787-826. doi: 10.1137/140972664.

Figure 1.  Performances of four methods on $T_1$-weighted brain MR images with different levels of noises
Figure 2.  Comparison among the four methods in terms of PSNR and MSSIM for images with different intensity inhomogeneities
Figure 3.  Performances of four methods on $T_1$-weighted brain images with different intensity inhomogeneities
Figure 4.  Ground truth and estimated bias field of the proposed method with examples in FIGURE 3
Figure 5.  Comparison of the performance in terms of CV(%)
Figure 6.  The relative errors of $r$ and $l$ and numerical energy of our model for the first image in FIGURE 3
Figure 7.  Tests on real MR images. The parameters used in our model are $\alpha = 0.03$ and $\beta = 0.015$
Figure 8.  Tests on color images of HoTVL1 model and our model
Figure 9.  Decomposition of the checkboard image
Figure 10.  Decomposition of the logvi image
Figure 11.  Denoising and decomposition of the images containing Poisson noise
Table 1.  PSNR and MSSIM of $T_1$-weighted brain MR images with different levels of noises
$3\%$ $5\%$ $7\%$ $9\%$
PSNR MSSIM PSNR MSSIM PSNR MSSIM PSNR MSSIM
Test Image1 TVH1 26.8369 0.9436 25.2879 0.9174 24.4622 0.8933 23.2509 0.8629
HoTVL1 29.7605 0.9515 27.4964 0.9342 26.6143 0.9214 25.2707 0.9063
L0MS 29.6193 0.9198 27.4895 0.9033 26.1466 0.8915 24.3937 0.8650
ETV 32.6749 0.9904 31.1170 0.9844 29.1291 0.9767 28.4244 0.9686
Test Image2 TVH1 27.3897 0.9229 26.4466 0.8932 25.4457 0.8655 23.7962 0.8334
HoTVL1 29.5698 0.9321 28.7948 0.9142 27.1248 0.8988 25.4362 0.8833
L0MS 31.8155 0.9227 28.5140 0.8950 27.1074 0.8746 24.8900 0.8374
ETV 33.4328 0.9911 31.4092 0.9842 29.8200 0.9764 28.6543 0.9698
$3\%$ $5\%$ $7\%$ $9\%$
PSNR MSSIM PSNR MSSIM PSNR MSSIM PSNR MSSIM
Test Image1 TVH1 26.8369 0.9436 25.2879 0.9174 24.4622 0.8933 23.2509 0.8629
HoTVL1 29.7605 0.9515 27.4964 0.9342 26.6143 0.9214 25.2707 0.9063
L0MS 29.6193 0.9198 27.4895 0.9033 26.1466 0.8915 24.3937 0.8650
ETV 32.6749 0.9904 31.1170 0.9844 29.1291 0.9767 28.4244 0.9686
Test Image2 TVH1 27.3897 0.9229 26.4466 0.8932 25.4457 0.8655 23.7962 0.8334
HoTVL1 29.5698 0.9321 28.7948 0.9142 27.1248 0.8988 25.4362 0.8833
L0MS 31.8155 0.9227 28.5140 0.8950 27.1074 0.8746 24.8900 0.8374
ETV 33.4328 0.9911 31.4092 0.9842 29.8200 0.9764 28.6543 0.9698
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